Over the last few months we have been looking at how we might approximate the solutions to ordinary differential equations, or ODEs, which define the derivative of one variable with respect to another with a function of them both. Firstly with the first order Euler method, then with the second order midpoint method and finally with the generalised Runge-Kutta method.

Unfortunately all of these approaches require the step length to be fixed and specified in advance, ignoring any information that we might use to adjust it during the iteration in order to better trade off the efficiency and accuracy of the approximation. In this post we shall try to automatically modify the step lengths to yield an optimal, or at least reasonable, balance.

Unfortunately all of these approaches require the step length to be fixed and specified in advance, ignoring any information that we might use to adjust it during the iteration in order to better trade off the efficiency and accuracy of the approximation. In this post we shall try to automatically modify the step lengths to yield an optimal, or at least reasonable, balance.

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